nLab functorial aspects of the GNS representation

Remark

This is a small subcollection of my (Tom Mainiero‘s) personal notes on some functorial aspects of the Gelfand-Naimark-Segal construction (GNS construction) that have been sitting around for a while. Please feel free to contribute! The ideas here underlie some of the constructs of Homtools: specifically, one can think of a “ltipartite’‘ state as a presheaf valued in the category of states (defined below) over a collection of subsystems. The GNS representation can then be thought of as a functor that takes us into a “representation-theoretic” category where we can, e.g. compute an associated (co)homology using Čech methods.

Baby’s first GNS functor

For the purposes of this section, $R$ is a fixed a unital ring and $Q$ is a unital $R$-algebra. Although $R$ will not appear explicitly in any of the discussion in this section (so that $Q$ could effectively just be thought of as a unital ring), we keep it around as the case where $Q$ is a finite-dimensional (semisimple) $\mathbb{C}$-algebra is most relevant to later discussions involving the GNS construction.

Definition

Let ${}_{Q}\mathbf{Mod}^{\bullet}$ denote the category with:

• objects given by pointed left $Q$-modules: pairs $(M,m)$ of a left $Q$-module $M$ with an element $m \in M$;

• morphisms given by point preserving intertwiners.

${}_{Q}\mathbf{Mod}^{\odot}$ is the full subcategory of ${}_{Q}\mathbf{Mod}^{\bullet}$ whose objects are cyclic pointed modules: $(M,m)$ such that $Q m = M$.

Remark

For those that find the notion of a pointed module somewhat awkward, note that: ${}_{Q}\mathbf{Mod}^{\bullet}$ is equivalent to the coslice category ${}_{Q}{Q}/{}_{Q}\mathbf{Mod}$, where ${}_{Q}\mathbf{Mod}$ is the category of left $Q$-modules.

Remark

${}_{Q}\mathbf{Mod}^{\odot}$ is a reflective subcategory of ${}_{Q}\mathbf{Mod}^{\bullet}$: the right adjoint of the inclusion functor

$$\iota \colon {}_{Q}\mathbf{Mod}^{\odot} \longrightarrow {}_{Q}\mathbf{Mod}^{\bullet}$$

is given by the “cyclification” functor:

$$\mathsf{Cyc} \colon {}_{Q}\mathbf{Mod}^{\bullet} \longrightarrow {}_{Q}\mathbf{Mod}^{\odot}$$

that acts on objects by assigning to any pointed module $(M,m)$ the cyclic pointed submodule generated by the orbit of $m$, i.e. $\mathsf{Cyc} \colon (M,m) \to (Q \cdot m, m)$ and acts on morphisms by restriction to cyclic submodules.

Notation

For any left $Q$-module $M$, we denote the annihilator of a point $m \in M$ as:

$$\mathrm{Ann}_{Q}(m) \coloneqq \{ q \in Q : q \cdot m = 0 \}.$$

The following lemma, which will become useful later, describes the structure of the hom-sets ${}_{Q}\mathbf{Mod}^{\bullet}((M,m), (N,n))$ when $(M,m)$ is a cyclic pointed module.

Lemma

Let $(M,m)$ and $(N,n)$ be pointed $Q$-modules and suppose $(M,m)$ is cyclic. Then there exists a morphism from $(M,m)$ to $(N,n)$ if and only if $\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n)$. Moreover, this morphism is unique when it exists.

Proof

This follows quickly from the fact that a morphism $f: (M,m) \rightarrow (N,n)$ must satisfy $f(q m) =q n$ for all $q \in Q$.

There is an immediate corollary of this Lemma: ${}_{Q}\mathbf{Mod}^{\odot}$ is equivalent to the poset of left-ideals of $Q$ (thought of as a category). This equivalence of categories is actually part of the restriction of a larger adjunction. We now spell out the definitions behind this claim.

Definition

$\mathbf{L}(Q)$ is the thin category given by the poset of left ideals of $Q$:

• objects are left ideals of $Q$, and

• there is a unique morphism $I \overset{!}{\to} J$ if and only if $I \subseteq J$.

The process of taking annihilators of distinguished points of left $Q$-modules defines a functor:

$$\mathsf{Ann}_{Q} \colon {}_{Q}\mathbf{Mod}^{\bullet} \longrightarrow \mathbf{L}(Q).$$

Explicitly: it acts on objects by:

$$\mathsf{Ann}_{Q} \colon (M,m) \mapsto \mathrm{Ann}_{Q}(m),$$

and its action on morphisms is deduced from the following straightforward remark.

Remark

If $f \colon (M,m) \to (N,n)$ is a morphism of pointed modules, then $q \cdot n = q \cdot f(m) = f(q \cdot m)$; so we must have $\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n)$.

Thus, for $f \colon (M,m) \to (N,n)$ a $\mathbf{Mod}^{\bullet}_{Q}$-morphism, we can define $\mathsf{Ann}_{Q}(f) \colon \mathsf{Ann}_{Q}(m) \to \mathsf{Ann}_{Q}(n)$ as the unique $\mathbf{L}(Q)$-morphism $\mathsf{Ann}_{Q}(m) \overset{!}{\to} \mathsf{Ann}_{Q}(n)$ given by the inclusion of ideals $\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n)$.

Alternatively, given any left-ideal $I$ of some $R$-algebra $Q$, one can produce an $Q$-module $Q/I$. This module has a distinguished point $[1_{Q}] \coloneqq 1_{Q} + I$, where $1_{Q}$ is the identity of $Q$. Moreover, $Q \cdot [1_{Q}] = Q/I$; so, so the pointed module $(Q/I, [1])$ is cyclic. This process defines a functor:

$$\mathsf{Quot} \colon \mathbf{L}(Q) \longrightarrow {}_{Q}\mathbf{Mod}^{\odot}$$

whose action on objects is as described above:

$$\mathsf{Quot} \colon I \mapsto (Q/I, [1_{Q}])$$

and action on morphisms is given by:

$$\mathsf{Quot} \colon (I \overset{!}{\to} J) \mapsto \tau^{I}_{J}$$

where $\tau^{I}_{J}$ is the tautological map of pointed modules acting by:

$$\tau^{I}_{J}: q + I \mapsto q + J$$

for any $q \in Q$; this is well-defined as $J \subseteq I$.

Proposition

There is an adjunction:

restricting to an equivalence

where $\widehat{\mathsf{Quot}}_{Q} = \iota \circ \mathsf{Quot}_{Q}$ for $\iota$ the inclusion functor of cyclic pointed modules into pointed modules.

Proof

We will define a unit

$$\eta \colon \mathrm{id}_{\mathbf{L}(Q)} \longrightarrow \mathsf{Ann}_{Q} \circ \mathsf{Quot}$$

and counit

$$\epsilon \colon \mathsf{Quot} \circ \mathsf{Ann}_{Q} \longrightarrow \mathrm{id}_{{}_{Q}\mathbf{Mod}^{\bullet}}.$$

satisfying the triangle identities. Note that $\eta$ is given by assigning the identity morphism to each object of $\mathbf{L}(Q)$: indeed, for any left-ideal we have $\mathsf{Ann}_{Q} \circ \mathsf{Quot}(I) = \mathsf{Ann}_{Q} \left[(Q/I,[1]) \right] = I$. The counit $\epsilon$ is a bit more interesting: let $\mathsf{M} = (M,m)$ be a pointed-module and define $(K,k) \coloneqq \mathsf{Quot} \circ \mathsf{Ann}_{Q}(\mathsf{M}) = (Q/\mathrm{Ann}_{Q}(m),[1])$. We have $\mathrm{Ann}_{Q}(k) = \mathrm{Ann}_{Q}(m)$; so by Lemma there is a unique morphism $(K,k) \rightarrow (M,m)$; this defines the morphism $\epsilon_{(M,m)}$. The triangle identities are trivial.

Connection to the GNS construction

The above observations are simple shadows of the functorality of the GNS construction for C*-algebras and W*-algebras. The opposite of the category of ideals $\mathbf{L}(Q)$ can be thought of as a stand-in for the category of states (to every $C^\ast$-algebraic state $\rho \colon E \to \mathbb{C}$ on a $C^{\ast}$-algebra $E$ we can associate the left ideal given by the “left kernel” $\{e \in E: \rho(e^{*} e) = 0 \}$). The quotient functor $\mathsf{Quot}$ is a toy-model of the GNS construction.

Notation/Terminology

In the following the letters $E, F$ and $G$ will denote $C^\ast$-algebras that are not necessarily $\W^\ast$-algebras and $A, B, C$ will denote $W^\ast$-algebras.

The term (normal) state will be taken to mean (normal) positive linear functional Note that this is at odds with some definitions in the physics literature, where “state” typically means “normalized” positive linear functional (if $f(1_{E}) = 1$ for $f: E \to \mathbb{C}$ a positive linear functional). Using the definition of “state” here, the zero map $0: E \to \mathbb{C}$ is a perfectly valid state.

The GNS construction as a functor on a fixed C$^\ast$-algebra

With an appropriate notion of a “category of states”, one can explore functorial properties of the GNS construction as in Arthur Parzygnat’s work (see Parzygnat2016 and Parzygnat2018).

The construction here is from a slightly different perspective, using a mildly richer category of states based on the natural poset structure on the collection of (normal) states on a fixed $W^\ast$ or $C^\ast$-algebra. The posetal relation is intimately related to the notion of a Radon-Nikodym derivative.

The category of (normal) states over a fixed (W$^\ast$) C$^\ast$-algebra.

Begin by recalling a well-known posetal relation on the set of states $\text{State(E)}$ on a $C^\ast$-algebra $E$:

Definition

For any $f, g \in \text{State}(E)$, define the relation $\leq$ by $f \leq g$ if $f - g$ is a state, where $-$ denotes pointwise subtraction; equivalently: $f \leq g$ if $f(e^\ast e) \leq g(e^\ast e)$ for all $e \in E$

It is a straightforward exercise to show this relation is posetal.

The story for normal states in the $W^\ast$-algebra world is nearly identical: let $\text{NState(A)}$ denote the set of normal states on a $W^\ast$-algebra $A$ (a subset of $\text{State}(A)$); for any $f, g \in \text{NState(A)}$ we can define a relation $\leq_{\text{norm}}$ by: $f \leq_{\text{norm}} g$ if $f - g$ is a normal state. Because the space of linear functionals splits as an orthogonal sum into normal and singular parts, it follows immediately that $f \leq_{\text{norm}} g$ for normal linear functionals $f$ and $g$ if and only if $f \leq g$. As a result we can simply write $\leq$ when discussing states on $C^\ast$-algebras or normal states on $W^\ast$-algebras without any issues if we forget down from the category of $W^\ast$-algebras to the category of $C^\ast$-algebras.

Remark

We can easily define weaker versions of the relation $\leq$ that may also be of interest: for $f$ and $g$ positive linear functionals on a $C^\ast$ algebra $E$, we can define relations $\ll$ and $\lesssim$ in the following manner:

• $f \ll g$ if there is an inclusion of $\ker^{L}(g) \subseteq \ker^{L}(f)$, where, for any positive linear functional $h$ on $E$, the “left kernel” or “vanishing ideal” is defined by:

  $$\ker^{L}(h) \coloneqq \{e \in E: h(e^\ast e ) = 0 \}.$$

• $f \lesssim g$ if there exists a $C \geq 0$ such that $f(e^\ast e) \leq C g(e^\ast e)$ for every $e \in E$.

We have the following immediate properties:

• $f \leq g \Rightarrow f \lesssim g \Rightarrow f \ll g$.

• $\leq$ is a partial order as it is antisymmetric: $f \leq g$ and $g \leq f$ if and only if $f = g$.

However, the weaker preorders $\ll$ and $\lesssim$ do not satisfy antisymmetry: e.g. $2 f \lesssim f$ and $f \lesssim 2f$.

Moreover, while $\lesssim$ is only a mild weakening of $\leq$, the weakest preorder $\ll$ is vastly different. In particular, we will show there is a unique Radon-Nikodym derivative associated to every pair of functionals related by the preorders $\leq$ and $\lesssim$; the result can be weakened for $\ll$, but the Radon-Nikodym derivative will be an unbounded (densely defined) operator. See Gudder.

Definition

Let $E$ be a $C^\ast$-algebra. The category $\mathbf{State}(E)$ is the opposite of the thin category determined by the preorder $\leq$, i.e. it is the category whose objects are all states (positive linear functionals) on $E$, and whose morphism sets are given by

$$\mathbf{State}(E)(\omega, \nu) = \left\{ \begin{array}{ll} \{\star\}, & \nu \leq \omega\\ \emptyset, & \text{otherwise} \end{array} \right.$$

where $\{\star\}$ denotes the one point set. Similarly if $A$ is a $W^\ast$-algebra, let $\mathbf{NState}(A)$ be defined as the full subcategory of $\mathbf{State}(A)$ whose objects are normal states.

Definition

For $E$ a $C^\ast$-algebra, the category $\mathbf{Rep}^{\bullet}(E)$ is the category with:

• objects given by pointed $\ast$-representations of $E$: triples $(\mathcal{H}, v, \pi: E \to \mathrm{B} \mathcal{H})$ of a Hilbert space $\mathcal{H}$ a vector $v \in \mathcal{H}$ and a $\ast$-representation $\pi$;

• morphisms given by contractive linear point-preserving intertwiners: a morphism from $(\mathcal{H}_{1}, v_{1}, \pi_1)$ to $(\mathcal{H}_2, v_2, \pi_2)$ is a linear map $f: \mathcal{H}_{1} \to \mathcal{H}_{2}$ such that $\|f\| \leq 1$ with respect to the operator norm, $f v_{1} = v_{2}$, and

  $$f \circ \pi_{1}(e) = \pi_{2}(e) \circ f$$

  for all $e \in E$.

$\mathbf{Rep}^{\odot}(E)$ is the full subcategory of $\mathbf{Rep}^{\bullet}(E)$ whose objects are cyclic representations: $(\mathcal{H}, v, \pi: E \to \mathrm{B} \mathcal{H})$ such that the subspace $E v = \{e v : e \in E \}$ is dense in $\mathcal{H}$.

We also define normal versions of the above.

Definition

Similarly, for $A$ a $W^\ast$-algebra, the category $\mathbf{NRep}^{\bullet}(A)$ is the category whose objects are pointed normal $\ast$-representations of $A$, and morphisms are contractive linear intertwiners (equivalently the full subcategory of $\mathbf{Rep}^{\bullet}(A)$ generated by objects whose underlying representation is normal). $\mathbf{NRep}^{\odot}(A)$ is the full subcategory of $\mathbf{NRep}^{\bullet}(A)$ generated by cyclic representations.

Maybe add a bit here about how NRep and Rep interact.

The following are a few straightforward observations about the subcategories $\mathbf{(N)Rep}^{\odot}(\ldots)$ generated by cyclic objects.

Remark

The cyclic subcategories are thin: there is at most one morphism between two objects, every morphism is uniquely determined by where distinguished point is sent.

Remark

Just as the full subcategory of cyclic pointed $Q$-modules (for $Q$ an $R$-algebra) is reflective inside of the category of pointed $A$-modules (c.f.\ Remark ), the subcategory $\mathbf{Rep}^{\odot}(E)$ (resp. $\mathbf{NRep^{\odot}}(A)$) is a reflective subcategories of $\mathbf{Rep}^{\bullet}(A)$ (resp. $\mathbf{NRep^{\bullet}}(A)$). The right adjoint of the inclusion functor is given on objects by assigning to any pointed representation the completion of the orbit of the distinguished point; it is defined on morphisms by restriction.

The observations in the above remark are shadows of adjunctions ($E$ a $C^{\ast}$-algebra and $A$ a $W^{\ast}$-algebra)

which restricts to an equivalence of categories:

The analogous statements hold for normal states/pointed representations. Our next step is to unravel the functors in the above diagrams, both which should be clear to those familiar with the GNS construction

Definition

The functor:

$$\mathsf{Fnl} \colon \mathbf{Rep}^{\bullet}(E) \longrightarrow \mathbf{State}(E)$$

acts on objects by:

$$\mathsf{Fnl} \colon (\mathcal{H}, v, \pi: E \rightarrow \operatorname{B} \mathcal{H}) \mapsto (e \mapsto \langle v, \pi(e) v \rangle^{\mathcal{H}} ),$$

and takes a morphism $f \in \mathbf{Rep}^{\bullet}(E)[(\mathcal{H}_{1}, v_{1}, \pi_{1}), (\mathcal{H}_{2}, v_{2}, \pi_{2})]$ to the unique morphism in $\mathbf{State}(E)$ defined by the relation:

$$\langle v_{2}, \pi_{2}( e^{*} e ) v_{2} \rangle = \langle f v_{1}, \pi_{2}( e^{*} e ) f v_{1} \rangle = \langle v_{1}, f^{*} f \pi_{1}( e^{*} e ) v_{1} \rangle \leq \langle v_{1}, \pi_{1}( e^{*} e) v_{1} \rangle,$$

for all $e \in E$.

Before defining the functor $\mathsf{L^2}$, we note that to any positive linear functional $\omega \colon E \to \mathbb{C}$, there is an associated left ideal $\ker^{L} \omega$. Thus, repeating the observations of Section, there is an associated pointed (algebraic) left $E$-module $E/\ker^{L} \omega$

Definition

The functor $\mathsf{L^2} \colon \mathbf{State}(E) \rightarrow \mathbf{Rep}^{\bullet}(E)$ acts on objects by taking a positive linear functional $\omega$ to its corresponding GNS representation, and acts on a morphism $\nu \to \omega$ corresponding the relation $\omega \leq \nu$ by assigning the quotient map $\overline{q}(\omega, \nu) \colon \mathsf{L^{2}} \nu \to \mathsf{L^{2}} \omega$

References

• Tom Mainiero, Homological Tools for the Quantum Mechanic (2019) [arXiv:1901.02011, 10.10.11.6/handle/1/1339]

• Arthur Parzygnat, From observables and states to Hilbert space and back: a 2-categorical adjunction, (arXiv:1609.08975)

• Arthur Parzygnat, Stinespring’s construction as an adjunction (arXiv:1807.02533)

• MathOverflow: In which sense the GNS construction is a functor? (link)

• Stanley Gudder?, A Radon-Nikodym Theorem for $\ast$-Algebras (link)